Copyright © University of Cambridge. All rights reserved.

## 'Proof Sorter - the Square Root of 2 Is Irrational' printed from http://nrich.maths.org/

Full screen
version
**If you can see this message Flash may not be working in your browser**
**Please see http://nrich.maths.org/techhelp/#flash to enable it.**
This method of proof can
easily be generalised to prove that $\sqrt n$ is irrational when
$n$ is not a square number .

What is the length of the
diagonal of a square with sides of length 2?

How do we find the value of
$\sqrt 2$?

What number has 2 as its
square?

What is the side of a square
which has area 2?

Now $(1.4)^2=1.96$, so the number $\sqrt 2$ is roughly $1.4$.
To get a better approximation divide $2$ by $1.4$ giving about
$1.428$, and take the average of $1.4$ and $1.428$ to get $1.414$.
Repeating this process, $2\div 1.414 \approx 1.41443$ so $2\approx
1.414 \times 1.41443$, and the average of these gives the next
approximation $1.414215$. We can continue this process indefinitely
getting better approximations but never finding the square root
exactly.

If $\sqrt 2$ were a rational number, that is if it could be
written as a fraction $p/q$ where $p$ and $q$ are integers, then we
could find the exact value. The proof sorter shows that this number
is IRRATIONAL so we cannot find an exact value.